// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_MATRIX_LOGARITHM
#define EIGEN_MATRIX_LOGARITHM

namespace Eigen {

namespace internal {

template<typename Scalar>
struct matrix_log_min_pade_degree
{
	static const int value = 3;
};

template<typename Scalar>
struct matrix_log_max_pade_degree
{
	typedef typename NumTraits<Scalar>::Real RealScalar;
	static const int value = std::numeric_limits<RealScalar>::digits <= 24 ? 5 : // single precision
								 std::numeric_limits<RealScalar>::digits <= 53 ? 7
																			   : // double precision
								 std::numeric_limits<RealScalar>::digits <= 64 ? 8
																			   : // extended precision
								 std::numeric_limits<RealScalar>::digits <= 106 ? 10
																				: // double-double
								 11;											  // quadruple precision
};

/** \brief Compute logarithm of 2x2 triangular matrix. */
template<typename MatrixType>
void
matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
{
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	using std::abs;
	using std::ceil;
	using std::imag;
	using std::log;

	Scalar logA00 = log(A(0, 0));
	Scalar logA11 = log(A(1, 1));

	result(0, 0) = logA00;
	result(1, 0) = Scalar(0);
	result(1, 1) = logA11;

	Scalar y = A(1, 1) - A(0, 0);
	if (y == Scalar(0)) {
		result(0, 1) = A(0, 1) / A(0, 0);
	} else if ((abs(A(0, 0)) < RealScalar(0.5) * abs(A(1, 1))) || (abs(A(0, 0)) > 2 * abs(A(1, 1)))) {
		result(0, 1) = A(0, 1) * (logA11 - logA00) / y;
	} else {
		// computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
		RealScalar unwindingNumber = ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2 * EIGEN_PI));
		result(0, 1) =
			A(0, 1) * (numext::log1p(y / A(0, 0)) + Scalar(0, RealScalar(2 * EIGEN_PI) * unwindingNumber)) / y;
	}
}

/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
inline int
matrix_log_get_pade_degree(float normTminusI)
{
	const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */,
									 4.0535837411880493e-1,
									 5.3149729967117310e-1 };
	const int minPadeDegree = matrix_log_min_pade_degree<float>::value;
	const int maxPadeDegree = matrix_log_max_pade_degree<float>::value;
	int degree = minPadeDegree;
	for (; degree <= maxPadeDegree; ++degree)
		if (normTminusI <= maxNormForPade[degree - minPadeDegree])
			break;
	return degree;
}

/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
inline int
matrix_log_get_pade_degree(double normTminusI)
{
	const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */,
									  5.3873532631381171e-2,
									  1.1352802267628681e-1,
									  1.8662860613541288e-1,
									  2.642960831111435e-1 };
	const int minPadeDegree = matrix_log_min_pade_degree<double>::value;
	const int maxPadeDegree = matrix_log_max_pade_degree<double>::value;
	int degree = minPadeDegree;
	for (; degree <= maxPadeDegree; ++degree)
		if (normTminusI <= maxNormForPade[degree - minPadeDegree])
			break;
	return degree;
}

/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
inline int
matrix_log_get_pade_degree(long double normTminusI)
{
#if LDBL_MANT_DIG == 53 // double precision
	const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */,
										   5.3873532631381171e-2L,
										   1.1352802267628681e-1L,
										   1.8662860613541288e-1L,
										   2.642960831111435e-1L };
#elif LDBL_MANT_DIG <= 64  // extended precision
	const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */,
										   2.34559162387971167321e-2L,
										   5.84603923897347449857e-2L,
										   1.08486423756725170223e-1L,
										   1.68385767881294446649e-1L,
										   2.32777776523703892094e-1L };
#elif LDBL_MANT_DIG <= 106 // double-double
	const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
										   9.34074328446359654039446552677759e-4L,
										   4.26117194647672175773064114582860e-3L,
										   1.21546224740281848743149666560464e-2L,
										   2.61100544998339436713088248557444e-2L,
										   4.66170074627052749243018566390567e-2L,
										   7.32585144444135027565872014932387e-2L,
										   1.05026503471351080481093652651105e-1L };
#else					   // quadruple precision
	const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
										   5.8853168473544560470387769480192666e-4L,
										   2.9216120366601315391789493628113520e-3L,
										   8.8415758124319434347116734705174308e-3L,
										   1.9850836029449446668518049562565291e-2L,
										   3.6688019729653446926585242192447447e-2L,
										   5.9290962294020186998954055264528393e-2L,
										   8.6998436081634343903250580992127677e-2L,
										   1.1880960220216759245467951592883642e-1L };
#endif
	const int minPadeDegree = matrix_log_min_pade_degree<long double>::value;
	const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value;
	int degree = minPadeDegree;
	for (; degree <= maxPadeDegree; ++degree)
		if (normTminusI <= maxNormForPade[degree - minPadeDegree])
			break;
	return degree;
}

/* \brief Compute Pade approximation to matrix logarithm */
template<typename MatrixType>
void
matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
{
	typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
	const int minPadeDegree = 3;
	const int maxPadeDegree = 11;
	assert(degree >= minPadeDegree && degree <= maxPadeDegree);
	// FIXME this creates float-conversion-warnings if these are enabled.
	// Either manually convert each value, or disable the warning locally
	const RealScalar nodes[][maxPadeDegree] = { { 0.1127016653792583114820734600217600L,
												  0.5000000000000000000000000000000000L, // degree 3
												  0.8872983346207416885179265399782400L },
												{ 0.0694318442029737123880267555535953L,
												  0.3300094782075718675986671204483777L, // degree 4
												  0.6699905217924281324013328795516223L,
												  0.9305681557970262876119732444464048L },
												{ 0.0469100770306680036011865608503035L,
												  0.2307653449471584544818427896498956L, // degree 5
												  0.5000000000000000000000000000000000L,
												  0.7692346550528415455181572103501044L,
												  0.9530899229693319963988134391496965L },
												{ 0.0337652428984239860938492227530027L,
												  0.1693953067668677431693002024900473L, // degree 6
												  0.3806904069584015456847491391596440L,
												  0.6193095930415984543152508608403560L,
												  0.8306046932331322568306997975099527L,
												  0.9662347571015760139061507772469973L },
												{ 0.0254460438286207377369051579760744L,
												  0.1292344072003027800680676133596058L, // degree 7
												  0.2970774243113014165466967939615193L,
												  0.5000000000000000000000000000000000L,
												  0.7029225756886985834533032060384807L,
												  0.8707655927996972199319323866403942L,
												  0.9745539561713792622630948420239256L },
												{ 0.0198550717512318841582195657152635L,
												  0.1016667612931866302042230317620848L, // degree 8
												  0.2372337950418355070911304754053768L,
												  0.4082826787521750975302619288199080L,
												  0.5917173212478249024697380711800920L,
												  0.7627662049581644929088695245946232L,
												  0.8983332387068133697957769682379152L,
												  0.9801449282487681158417804342847365L },
												{ 0.0159198802461869550822118985481636L,
												  0.0819844463366821028502851059651326L, // degree 9
												  0.1933142836497048013456489803292629L,
												  0.3378732882980955354807309926783317L,
												  0.5000000000000000000000000000000000L,
												  0.6621267117019044645192690073216683L,
												  0.8066857163502951986543510196707371L,
												  0.9180155536633178971497148940348674L,
												  0.9840801197538130449177881014518364L },
												{ 0.0130467357414141399610179939577740L,
												  0.0674683166555077446339516557882535L, // degree 10
												  0.1602952158504877968828363174425632L,
												  0.2833023029353764046003670284171079L,
												  0.4255628305091843945575869994351400L,
												  0.5744371694908156054424130005648600L,
												  0.7166976970646235953996329715828921L,
												  0.8397047841495122031171636825574368L,
												  0.9325316833444922553660483442117465L,
												  0.9869532642585858600389820060422260L },
												{ 0.0108856709269715035980309994385713L,
												  0.0564687001159523504624211153480364L, // degree 11
												  0.1349239972129753379532918739844233L,
												  0.2404519353965940920371371652706952L,
												  0.3652284220238275138342340072995692L,
												  0.5000000000000000000000000000000000L,
												  0.6347715779761724861657659927004308L,
												  0.7595480646034059079628628347293048L,
												  0.8650760027870246620467081260155767L,
												  0.9435312998840476495375788846519636L,
												  0.9891143290730284964019690005614287L } };

	const RealScalar weights[][maxPadeDegree] = { { 0.2777777777777777777777777777777778L,
													0.4444444444444444444444444444444444L, // degree 3
													0.2777777777777777777777777777777778L },
												  { 0.1739274225687269286865319746109997L,
													0.3260725774312730713134680253890003L, // degree 4
													0.3260725774312730713134680253890003L,
													0.1739274225687269286865319746109997L },
												  { 0.1184634425280945437571320203599587L,
													0.2393143352496832340206457574178191L, // degree 5
													0.2844444444444444444444444444444444L,
													0.2393143352496832340206457574178191L,
													0.1184634425280945437571320203599587L },
												  { 0.0856622461895851725201480710863665L,
													0.1803807865240693037849167569188581L, // degree 6
													0.2339569672863455236949351719947755L,
													0.2339569672863455236949351719947755L,
													0.1803807865240693037849167569188581L,
													0.0856622461895851725201480710863665L },
												  { 0.0647424830844348466353057163395410L,
													0.1398526957446383339507338857118898L, // degree 7
													0.1909150252525594724751848877444876L,
													0.2089795918367346938775510204081633L,
													0.1909150252525594724751848877444876L,
													0.1398526957446383339507338857118898L,
													0.0647424830844348466353057163395410L },
												  { 0.0506142681451881295762656771549811L,
													0.1111905172266872352721779972131204L, // degree 8
													0.1568533229389436436689811009933007L,
													0.1813418916891809914825752246385978L,
													0.1813418916891809914825752246385978L,
													0.1568533229389436436689811009933007L,
													0.1111905172266872352721779972131204L,
													0.0506142681451881295762656771549811L },
												  { 0.0406371941807872059859460790552618L,
													0.0903240803474287020292360156214564L, // degree 9
													0.1303053482014677311593714347093164L,
													0.1561735385200014200343152032922218L,
													0.1651196775006298815822625346434870L,
													0.1561735385200014200343152032922218L,
													0.1303053482014677311593714347093164L,
													0.0903240803474287020292360156214564L,
													0.0406371941807872059859460790552618L },
												  { 0.0333356721543440687967844049466659L,
													0.0747256745752902965728881698288487L, // degree 10
													0.1095431812579910219977674671140816L,
													0.1346333596549981775456134607847347L,
													0.1477621123573764350869464973256692L,
													0.1477621123573764350869464973256692L,
													0.1346333596549981775456134607847347L,
													0.1095431812579910219977674671140816L,
													0.0747256745752902965728881698288487L,
													0.0333356721543440687967844049466659L },
												  { 0.0278342835580868332413768602212743L,
													0.0627901847324523123173471496119701L, // degree 11
													0.0931451054638671257130488207158280L,
													0.1165968822959952399592618524215876L,
													0.1314022722551233310903444349452546L,
													0.1364625433889503153572417641681711L,
													0.1314022722551233310903444349452546L,
													0.1165968822959952399592618524215876L,
													0.0931451054638671257130488207158280L,
													0.0627901847324523123173471496119701L,
													0.0278342835580868332413768602212743L } };

	MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
	result.setZero(T.rows(), T.rows());
	for (int k = 0; k < degree; ++k) {
		RealScalar weight = weights[degree - minPadeDegree][k];
		RealScalar node = nodes[degree - minPadeDegree][k];
		result +=
			weight *
			(MatrixType::Identity(T.rows(), T.rows()) + node * TminusI).template triangularView<Upper>().solve(TminusI);
	}
}

/** \brief Compute logarithm of triangular matrices with size > 2.
 * \details This uses a inverse scale-and-square algorithm. */
template<typename MatrixType>
void
matrix_log_compute_big(const MatrixType& A, MatrixType& result)
{
	typedef typename MatrixType::Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	using std::pow;

	int numberOfSquareRoots = 0;
	int numberOfExtraSquareRoots = 0;
	int degree;
	MatrixType T = A, sqrtT;

	const int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value;
	const RealScalar maxNormForPade = RealScalar(maxPadeDegree <= 5 ? 5.3149729967117310e-1L : // single precision
													 maxPadeDegree <= 7 ? 2.6429608311114350e-1L
																		: // double precision
													 maxPadeDegree <= 8 ? 2.32777776523703892094e-1L
																		: // extended precision
													 maxPadeDegree <= 10 ? 1.05026503471351080481093652651105e-1L
																		 :						// double-double
													 1.1880960220216759245467951592883642e-1L); // quadruple precision

	while (true) {
		RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
		if (normTminusI < maxNormForPade) {
			degree = matrix_log_get_pade_degree(normTminusI);
			int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
			if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
				break;
			++numberOfExtraSquareRoots;
		}
		matrix_sqrt_triangular(T, sqrtT);
		T = sqrtT.template triangularView<Upper>();
		++numberOfSquareRoots;
	}

	matrix_log_compute_pade(result, T, degree);
	result *= pow(RealScalar(2), RealScalar(numberOfSquareRoots)); // TODO replace by bitshift if possible
}

/** \ingroup MatrixFunctions_Module
 * \class MatrixLogarithmAtomic
 * \brief Helper class for computing matrix logarithm of atomic matrices.
 *
 * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other.
 *
 * \sa class MatrixFunctionAtomic, MatrixBase::log()
 */
template<typename MatrixType>
class MatrixLogarithmAtomic
{
  public:
	/** \brief Compute matrix logarithm of atomic matrix
	 * \param[in]  A  argument of matrix logarithm, should be upper triangular and atomic
	 * \returns  The logarithm of \p A.
	 */
	MatrixType compute(const MatrixType& A);
};

template<typename MatrixType>
MatrixType
MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
{
	using std::log;
	MatrixType result(A.rows(), A.rows());
	if (A.rows() == 1)
		result(0, 0) = log(A(0, 0));
	else if (A.rows() == 2)
		matrix_log_compute_2x2(A, result);
	else
		matrix_log_compute_big(A, result);
	return result;
}

} // end of namespace internal

/** \ingroup MatrixFunctions_Module
 *
 * \brief Proxy for the matrix logarithm of some matrix (expression).
 *
 * \tparam Derived  Type of the argument to the matrix function.
 *
 * This class holds the argument to the matrix function until it is
 * assigned or evaluated for some other reason (so the argument
 * should not be changed in the meantime). It is the return type of
 * MatrixBase::log() and most of the time this is the only way it
 * is used.
 */
template<typename Derived>
class MatrixLogarithmReturnValue : public ReturnByValue<MatrixLogarithmReturnValue<Derived>>
{
  public:
	typedef typename Derived::Scalar Scalar;
	typedef typename Derived::Index Index;

  protected:
	typedef typename internal::ref_selector<Derived>::type DerivedNested;

  public:
	/** \brief Constructor.
	 *
	 * \param[in]  A  %Matrix (expression) forming the argument of the matrix logarithm.
	 */
	explicit MatrixLogarithmReturnValue(const Derived& A)
		: m_A(A)
	{
	}

	/** \brief Compute the matrix logarithm.
	 *
	 * \param[out]  result  Logarithm of \c A, where \c A is as specified in the constructor.
	 */
	template<typename ResultType>
	inline void evalTo(ResultType& result) const
	{
		typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
		typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
		typedef internal::traits<DerivedEvalTypeClean> Traits;
		typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
		typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, Traits::RowsAtCompileTime, Traits::ColsAtCompileTime>
			DynMatrixType;
		typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
		AtomicType atomic;

		internal::matrix_function_compute<typename DerivedEvalTypeClean::PlainObject>::run(m_A, atomic, result);
	}

	Index rows() const { return m_A.rows(); }
	Index cols() const { return m_A.cols(); }

  private:
	const DerivedNested m_A;
};

namespace internal {
template<typename Derived>
struct traits<MatrixLogarithmReturnValue<Derived>>
{
	typedef typename Derived::PlainObject ReturnType;
};
}

/********** MatrixBase method **********/

template<typename Derived>
const MatrixLogarithmReturnValue<Derived>
MatrixBase<Derived>::log() const
{
	eigen_assert(rows() == cols());
	return MatrixLogarithmReturnValue<Derived>(derived());
}

} // end namespace Eigen

#endif // EIGEN_MATRIX_LOGARITHM
